Spherically symmetric spacetime

In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially-moving dust, compressible and incompressible fluids (such as dark matter) or baryons (hydrogen). Because spherically symmetric spacetimes are by definition irrotational, they are not realistic models of black holes in nature; however, the sphere symmetry allows a metric of a considerably simpler form than that of a rotating spacetime, making sphere-symmetric problems much easier to solve. Such models are not entirely inappropriate: they often have a Penrose diagram similar to a rotating spacetime, and so typically have qualitative features (such as Cauchy horizons) that carry on to rotating spacetimes. One such application is the study of mass inflation due to counter-moving streams of infalling matter in the interior of a black hole. In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially-moving dust, compressible and incompressible fluids (such as dark matter) or baryons (hydrogen). Because spherically symmetric spacetimes are by definition irrotational, they are not realistic models of black holes in nature; however, the sphere symmetry allows a metric of a considerably simpler form than that of a rotating spacetime, making sphere-symmetric problems much easier to solve. Such models are not entirely inappropriate: they often have a Penrose diagram similar to a rotating spacetime, and so typically have qualitative features (such as Cauchy horizons) that carry on to rotating spacetimes. One such application is the study of mass inflation due to counter-moving streams of infalling matter in the interior of a black hole. A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the rotation group SO(3) and the orbits of this group are 2-spheres (ordinary 2-dimensional spheres in 3-dimensional Euclidean space). The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one whose metric is 'invariant under rotations'. The spacetime metric induces a metric on each orbit 2-sphere (and this induced metric must be a multiple of the metric of a 2-sphere). Conventionally, the metric on the 2-sphere is written in polar coordinates as